Question: $\dfrac{ 8d + 10e }{ 7 } = \dfrac{ 10d + 7f }{ 5 }$ Solve for $d$.
Multiply both sides by the left denominator. $\dfrac{ 8d + 10e }{ {7} } = \dfrac{ 10d + 7f }{ 5 }$ ${7} \cdot \dfrac{ 8d + 10e }{ {7} } = {7} \cdot \dfrac{ 10d + 7f }{ 5 }$ $8d + 10e = {7} \cdot \dfrac { 10d + 7f }{ 5 }$ Multiply both sides by the right denominator. $8d + 10e = 7 \cdot \dfrac{ 10d + 7f }{ {5} }$ ${5} \cdot \left( 8d + 10e \right) = {5} \cdot 7 \cdot \dfrac{ 10d + 7f }{ {5} }$ ${5} \cdot \left( 8d + 10e \right) = 7 \cdot \left( 10d + 7f \right)$ Distribute both sides ${5} \cdot \left( 8d + 10e \right) = {7} \cdot \left( 10d + 7f \right)$ ${40}d + {50}e = {70}d + {49}f$ Combine $d$ terms on the left. ${40d} + 50e = {70d} + 49f$ $-{30d} + 50e = 49f$ Move the $e$ term to the right. $-30d + {50e} = 49f$ $-30d = 49f - {50e}$ Isolate $d$ by dividing both sides by its coefficient. $-{30}d = 49f - 50e$ $d = \dfrac{ 49f - 50e }{ -{30} }$ Swap signs so the denominator isn't negative. $d = \dfrac{ -{49}f + {50}e }{ {30} }$